A truel is a three-way duel. The three participants take turns firing one shot at whichever opponent they choose, until only one is remaining.

Allison, Ben, and Chase are in a truel, and have varying degrees of accuracy: Allison has a 50% chance of hitting her intended target, Ben has a 80% chance, and Chase has a 100% chance. Allison gets to shoot first, then Ben, then Chase, and repeating in that order until only one person is remaining.

Assume that the accuracy of all participants are publicly known, and everyone is trying to maximize their chances of winning. What is Allison’s optimal strategy, and what is her likelihood of winning under that strategy?

Solution

Here is the logic (without showing all of the math):

• If Allison successfully hits Ben or Chase, she is left in a duel, but the other person gets to shoot first.
• If she hits Ben, then she as a 0% chance of winning, because Chase definitely hits her.
• If she hits Chase, then she has a 1/9 chance of winning:  P(Allison wins, when Ben and Allison are left and Ben shoots first) = P(Ben misses) x P(Allison hits) + P(Ben misses) x P(Allison misses) x P(Allison wins, when Ben and Allison are left and Ben shoots first)
• If she misses in either of the scenarios in #1, it’s easy to show that Chase will aim at Ben if it gets to Chase’s turn, because that would give Chase a 50% chance of winning instead of a 20% chance. Thus Ben will want to aim at Chase on his turn (otherwise Ben would definitely lose on Chase’s turn).
• So if she misses, 80% of the time she faces Ben, and 20% of the time she faces Chase. But now, she gets to shoot first in the resulting duel.
• This duel with Ben gives her a 5/9 chance of winning (use similar math as in 1b).
• This duel with Chase gives her a 1/2 chance of winning.
• Overall, that means she has a 49/90 (or 0.544) chance of winning if she misses her first shot.
• Thus (with some more math) if she aims at Chase, she has 59/180 chance of winning, and if she aims at Ben, she has a 49/180 chance. But we see that Allison actually maximizes her chances of winning if she aims at nothing/the ground/the sky (i.e. intentionally misses), giving her a 49/90 chance of winning.

So although Allison has the worst aim, she has the highest chance of winning – even higher than her chance of hitting anyone in the first place!